The minimum linear arrangement problem (MLA) consists of finding a mapping $\pi$ from vertices of a graph to integers that minimizes $\sum_{uv\in E}|\pi(u) - \pi(v)|$. For trees, various algorithms are available to solve the problem in polynomial time; the best known runs in subquadratic time in $n=|V|$. There exist variants of the MLA in which the arrangements are constrained to certain classes of projectivity. Iordanskii, and later Hochberg and Stallmann (HS), put forward $O(n)$-time algorithms that solve the problem when arrangements are constrained to be planar. We also consider linear arrangements of rooted trees that are constrained to be projective. Gildea and Temperley (GT) sketched an algorithm for the projectivity constraint which, as they claimed, runs in $O(n)$ but did not provide any justification of its cost. In contrast, Park and Levy claimed that GT's algorithm runs in $O(n \log d_{max})$ where $d_{max}$ is the maximum degree but did not provide sufficient detail. Here we correct an error in HS's algorithm for the planar case, show its relationship with the projective case, and derive an algorithm for the projective case that runs undoubtlessly in $O(n)$-time.
翻译:最起码的线性安排问题(MLA) 包括从图的顶端到整数的映射 $\ pi$, 使E ⁇ pi(u) -\pi(v)\\\$) $ 。 对于树木来说, 各种算法都可以在多元时间里解决问题; 最著名的在亚赤道时间里以$ ⁇ V ⁇ $ 美元进行。 存在将安排限制于某些种类的投影性的司法协助变量。 Iordanskii, 以及后来的Hoghberg和Stallmann(HS), 提出了美元(n) 美元- 美元- 时间算法, 在安排受约束时, 将问题解决到最小的 $- squax 。 我们还考虑对根树的线性安排进行预测; Gildea 和 Temperley (GT) 绘制了投影限制的算法, 正如他们所说的, 美元(n) 美元, 但没有提供其成本的任何理由。 (Park 和 Levy (Hn) $(nlog) rudeal) rial- developmental) ex case.